WAVE MOTIONS IN THE OCEAN

presented to

Myrl C. Hendershott

from

David C. Chapman and Paola Malanotte-Rizzoli

August, 1989

Placed on the web with Paola Malanotte-Rizzoli's permission




Contents


1 Basic concepts 1

1.1 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 The dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Linear superposition of plane waves . . . . . . . . . . . . . . . . . . . . . 6

1.4 The method of stationary phase: Group velocity . . . . . . . . . . . . . . 7

1.5 Waves in slowly varying media: Ray theory . . . . . . . . . . . . . . . . . 11


2 Acoustic waves 18

2.1 Basic physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Reflection at a solid boundary . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Plane waves in a channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Scattering at a discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 Generation of plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6.1 An initial value problem . . . . . . . . . . . . . . . . . . . . . . . 29

2.6.2 Forcing from rest . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.7 Slowly varying medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33


3 Surface gravity waves 38

3.1 Homogeneous medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Linear solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Internal waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Qualitative retreatment of surface waves . . . . . . . . . . . . . . . . . . 45

3.5 Careful retreatment of surface waves . . . . . . . . . . . . . . . . . . . . 47

3.6 An initial value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.7 Ship waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.8 A wave energy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.9 Slowly varying medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.10 Waves riding on a current . . . . . . . . . . . . . . . . . . . . . . . . . . 61


4 Internal gravity waves 64

4.1 The internal wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2 Unbounded, rotating. stratified fluid . . . . . . . . . . . . . . . . . . . . 68

4.3 Waveguide modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3.1 Evanescent modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4 Generation at a horizontal boundary . . . . . . . . . . . . . . . . . . . . 84

4.5 Reflection from a solid boundary . . . . . . . . . . . . . . . . . . . . . . 87

4.6 Variable buoyancy frequency . . . . . . . . . . . . . . . . . . . . . . . . . 92


5 Shallow water dynamics 96

5.1 Laplace's tidal equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2 Shallow water equations with rotation . . . . . . . . . . . . . . . . . . . 102

5.3 Reflection at a solid wall . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.4 Seiches in a box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.5 Propagation over a step . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.6 Edge waves and coastal seiches . . . . . . . . . . . . . . . . . . . . . . . 114

5.7 Sverdrup and Poincare waves . . . . . . . . . . . . . . . . . . . . . . . . 118

5.5 Kelvin waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.9 Waveguide modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.10 Kelvin wave reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.11 Rossby and planetary waves . . . . . . . . . . . . . . . . . . . . . . . . .1 27

5.12 Rossby wave reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.13 Western boundary current formation . . . . . . . . . . . . . . . . . . . . 139

5.14 Equatorial waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .140


6 Topographic effects 149

6.1 Topographic Rossby waves . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.2 Bottom-trapped waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.3 Continental shelf waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.4 Coastal-trapped waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.5 Wind-forced, long waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 167


7 References 172


Preface

When I volunteered to teach the MIT/WHOI Joint Program core course on

"Wave Motions in the Ocean and Atmosphere" in Spring 1989, I naturally turned for

guidance to the notes I had acquired from a similar course taken while a student at

Scripps Institution of Oceanography. In an attempt to broaden the scope of the course,

I borrowed a set of notes from Paola Malanotte-Rizzoli who taught the MIT/WHOI

core course from 1983-1985. It didn't take long to recognize that Paola's notes were

nearly identical to mine because she had also based hers on the waves course she had

taken at Scripps. In both cases, the Scripps course was taught by our former advisor

Myrl Hendershott which means that at least two generations of Physical

Oceanography students have learned the "Hendershott view" of waves. Considering

the seemingly timeless nature of the concepts presented in Myrl's course as well as the

profound influence Myrl has had on Paola and myself through both his teaching and

his advising, we decided to compile these notes into a form which could be distributed

to students and, at the same time, serve as a tribute to Myrl. Thus, with the exception

of some minor modifications, additions and deletions that Paola and I have made, the

notes contained herein are those developed by Myrl for his course. We hope that these

notes will be as clear and as useful to future readers as they have been to us.


Woods Hole

1989

David C. Chapman


These notes have been collected and assembled in different ways over the years

by two people successively, Paola Malanotte-Rizzoli and Dave Chapman. The present

and chronologically latest version has been put together by Dave and constitutes the

bulk of the waves course he taught in Spring 1989. When I taught the course during

the years 1983-85, the chapter on acoustic waves was absent. I had instead a section on

the Garrett and Munk spectrum and a chapter on nonlinear wave interactions. These

differences reflect the different years in which Dave and I took the waves course at

Scripps Institution of Oceanography from our former advisor Professor Myrl

Hendershott and the modifications that Myrl had made in his course in successive

years. Thus the inspirational source or, rather, the actual bulk of these notes is the

waves course taught by Myrl at Scripps.


Myrl Hendershott has been at WHOI this summer as Principal Lecturer of

the GFD summer School on Ocean circulation. This opportunity, plus Dave

Chapman's diligence and patience in typing the notes on his word processor together

with formulas and equations (the latter were handwritten in my own set of notes), has

motivated us to produce this report as an homage to Myrl. Without him, we would

both have had a much harder and more time-consuming role in putting together a

decent course on waves. More importantly, Myrl is in many ways responsible for

whatever success we have had in the field of Oceanography.

I must add here a personal note. Hearing Myrl again as a teacher this summer

after so many years, I have realized how much he has influenced My way of thinking

and teaching. On the not-so-positive side (I will not say negative): like him, I "scribble" a lot on the blackboard.

like him, I erase with my left hand what I have just written with my right hand.

like him, I put l (x wavenurnber) before k (y wavenumber)
As the letters
j, k, z, y, w do not exist in the Italian alphabet, x coming before y after

l was supremely unimportant to me. On the positive side, Myrl was absolutely the

best teacher I had in the various courses I took at Scripps. His lectures were always

interesting, imaginative and full of physical insight. Looking back, I realize that a

great deal of the important oceanographic concepts and ideas I learned over the years

go back to my long association with Myrl as teacher, advisor, colleague and, last but
not least, dear friend.
I hope I absorbed from him some of the positive qualities too.


Woods Hole

1989

Paola Malanotte- Rizzoli